MAE 3241: AERODYNAMICS ANDFLIGHT MECHANICS

Thrust and Power Requirements

April 28, 2010

Mechanical and Aerospace Engineering Department

Florida Institute of Technology

D. R. Kirk

2

EXAMPLE: BEECHCRAFT QUEEN AIR• The results we have developed so far for lift and drag for a finite wing may also be applied to a

complete airplane. In such relations:

– CD is drag coefficient for complete airplane

– CD,0 is parasitic drag coefficient, which contains not only profile drag of wing (cd) but also friction and pressure drag of tail surfaces, fuselage, engine nacelles, landing gear and any other components of airplane exposed to air flow

– CL is total lift coefficient, including small contributions from horizontal tail and fuselage

– Span efficiency for finite wing replaced with Oswald efficiency factor for entire airplane

• Example: To see how this works, assume the aerodynamicists have provided all the information needed about the complete airplane shown below

Beechcraft Queen Air Aircraft Data

W = 38,220 NS = 27.3 m2

AR = 7.5e (complete airplane) = 0.9CD,0 (complete airplane) = 0.03

What thrust and power levels are required of engines to cruise at 220 MPH at sea-level?How would these results change at 15,000 ft

3

OVERALL AIRPLANE DRAG• No longer concerned with aerodynamic details

• Drag for complete airplane, not just wing

eAR

CCC LDD

2

0, eAR

CCC LdD

2

Wing or airfoil Entire Airplane

Landing Gear

Engine Nacelles Tail Surfaces

4

DRAG POLAR• CD,0 is parasite drag coefficient at zero lift (L=0)

• CD,i drag coefficient due to lift (induced drag)

• Oswald efficiency factor, e, includes all effects from airplane

• CD,0 and e are known aerodynamics quantities of airplane

iDDL

DD CCeAR

CCC ,0,

2

0,

Example of Drag Polar for complete airplane

5

4 FORCES ACTING ON AIRPLANE• Model airplane as rigid body with four natural forces acting on it

1. Lift, L

• Acts perpendicular to flight path (always perpendicular to relative wind)

2. Drag, D

• Acts parallel to flight path direction (parallel to incoming relative wind)

3. Propulsive Thrust, T

• For most airplanes propulsive thrust acts in flight path direction

• May be inclined with respect to flight path angle, T, usually small angle

4. Weight, W

• Always acts vertically toward center of earth

• Inclined at angle, , with respect to lift direction

• Apply Newton’s Second Law (F=ma) to curvilinear flight path

– Force balance in direction parallel to flight path

– Force balance in direction perpendicular to flight path

6

GENERAL EQUATIONS OF MOTION (6.2)

cTlarperpendicu

Tparallel

r

VmWTLF

dt

dVmWDTF

2

cossin

sincos

Apply Newton’s 2nd Parallel to flight path:

Apply Newton’s 2nd Perpendicular to flight path:

Free Body Diagram

7

LEVEL, UNACCELERATED FLIGHT

• JSF is flying at constant speed (no accelerations)

• Sum of forces = 0 in two perpendicular directions

• Entire weight of airplane is perfectly balanced by lift (L = W)

• Engines produce just enough thrust to balance total drag at this airspeed (T = D)

• For most conventional airplanes T is small enough such that cos(T) ~ 1

T D

L

W

8

LEVEL, UNACCELERATED FLIGHT

L

D

L

D

C

C

W

T

SCVWL

SCVDT

WL

DT

2

2

2

12

1

DLW

CCW

T

D

LR

• TR is thrust required to fly at a given velocity in level, unaccelerated flight

• Notice that minimum TR is when airplane is at maximum L/D

– L/D is an important aero-performance quantity

9

THRUST REQUIREMENT (6.3)• TR for airplane at given altitude varies with velocity

• Thrust required curve: TR vs. V∞

10

PROCEDURE: THRUST REQUIREMENT1. Select a flight speed, V∞

2. Calculate CL

SV

WCL

2

21

eAR

CCC LDD

2

0,

D

LR

CCW

T

3. Calculate CD

4. Calculate CL/CD

5. Calculate TR

This is how much thrust enginemust produce to fly at selected V∞

Recall Homework Problem #5.6, find (L/D)max for NACA 2412 airfoil

Minimum TR when airplaneflying at (L/D)max

11

THRUST REQUIREMENT (6.3)• Different points on TR curve correspond to different angles of attack

eAR

CCSqSCqD

SCqSCVWL

LDD

LL

2

0,

2

2

1

At a:Large q∞

Small CL and D large

At b:Small q∞

Large CL (or CL2) and to support W

D large

12

THRUST REQUIRED VS. FLIGHT VELOCITY

eAR

CSqSCqT

CCSqSCqDT

LDR

iDDDR

2

0,

,0,

Zero-Lift TR

(Parasitic Drag)Lift-Induced TR

(Induced Drag)

Zero-Lift TR ~ V2

(Parasitic Drag)

Lift-Induced TR ~ 1/V2

(Induced Drag)

13

THRUST REQUIRED VS. FLIGHT VELOCITY

iDL

D

DR

RR

DR

CeAR

CC

eARSq

WSC

dq

dT

dq

dV

dV

dT

dq

dT

eARSq

WSCqT

,

2

0,

2

2

0,

2

0,

0

At point of minimum TR, dTR/dV∞=0

(or dTR/dq∞=0)

CD,0 = CD,i at minimum TR and maximum L/DZero-Lift Drag = Induced Drag at minimum TR and maximum L/D

14

HOW FAST CAN YOU FLY?• Maximum flight speed occurs when thrust available, TA=TR

– Reduced throttle settings, TR < TA

– Cannot physically achieve more thrust than TA which engine can provide

Intersection of TR

curve and maximumTA defined maximumflight speed of airplane

15

FURTHER IMPLICATIONS FOR DESIGN: VMAX

• Maximum velocity at a given altitude is important specification for new airplane

• To design airplane for given Vmax, what are most important design parameters?

21

0,

0,2

maxmaxmax

2

0,2

2

0,22

2

0,

2

0,

4

0

D

DAA

D

DD

L

LDD

C

eAR

C

WT

SW

SW

WT

V

eARS

WTqSCq

eARSq

WSCq

eARSq

WCSqT

Sq

WC

eAR

CCSqSCqTD

Steady, level flight: T = D

Steady, level flight: L = W

Substitute into drag equation

Turn this equation into a quadraticequation (by multiplying by q∞)and rearranging

Solve quadratic equation and setthrust, T, to maximum availablethrust, TA,max

16

FURTHER IMPLICATIONS FOR DESIGN: VMAX

• TA,max does not appear alone, but only in ratio (TA/W)max

• S does not appear alone, but only in ratio (W/S)

• Vmax does not depend on thrust alone or weight alone, but rather on ratios

– (TA/W)max: maximum thrust-to-weight ratio

– W/S: wing loading

• Vmax also depends on density (altitude), CD,0, eAR

• Increase Vmax by

– Increase maximum thrust-to-weight ratio, (TA/W)max

– Increasing wing loading, (W/S)

– Decreasing zero-lift drag coefficient, CD,0

21

0,

0,2

maxmaxmax

4

D

DAA

C

eAR

C

WT

SW

SW

WT

V

17

AIRPLANE POWER PLANTS

Two types of engines common in aviation today

1. Reciprocating piston engine with propeller

– Average light-weight, general aviation aircraft

– Rated in terms of POWER

2. Jet (Turbojet, turbofan) engine

– Large commercial transports and military aircraft

– Rated in terms of THRUST

18

THRUST VS. POWER• Jets Engines (turbojets, turbofans for military and commercial applications) are

usually rate in Thrust

– Thrust is a Force with units (N = kg m/s2)

– For example, the PW4000-112 is rated at 98,000 lb of thrust

• Piston-Driven Engines are usually rated in terms of Power

– Power is a precise term and can be expressed as:

• Energy / time with units (kg m2/s2) / s = kg m2/s3 = Watts

– Note that Energy is expressed in Joules = kg m2/s2

• Force * Velocity with units (kg m/s2) * (m/s) = kg m2/s3 = Watts

– Usually rated in terms of horsepower (1 hp = 550 ft lb/s = 746 W)

• Example:

– Airplane is level, unaccelerated flight at a given altitude with speed V∞

– Power Required, PR=TR*V∞

– [W] = [N] * [m/s]

19

POWER AVAILABLE (6.6)

Jet EnginePropeller Drive Engine

20

POWER AVAILABLE (6.6)

Jet EnginePropeller Drive Engine

21

POWER REQUIRED (6.5)

PR vs. V∞ qualitatively

(Resembles TR vs. V∞)

22

POWER REQUIRED (6.5)

D

LL

DR

L

D

LR

LL

D

LRR

CCSC

CWP

SC

W

CCW

P

SC

WVSCVWL

V

CCW

VTP

233

23

2

12

2

2

2

1

PR varies inversely as CL3/2/CD

Recall: TR varies inversely as CL/CD

23

POWER REQUIRED (6.5)

eAR

CSVqVSCqP

VCCSqVSCqDVVTP

LDR

iDDDRR

2

0,

,0,

Zero-Lift PR Lift-Induced PR

Zero-Lift PR ~ V3

Lift-Induced PR ~ 1/V

24

POWER REQUIRED

03

1

2

3212

1

,0,2

2

0,3

iDDR

DR

CCSVdV

dP

eARSV

WSCVP

iDD CC ,0, 3

1

At point of minimum PR, dPR/dV∞=0

25

POWER REQUIRED• V∞ for minimum PR is less than V∞ for minimum TR

iDD CC ,0,

iDD CC ,0, 3

1

26

WHY DO WE CARE ABOUT THIS?• We will show that for a piston-engine propeller combination

– To fly longest distance (maximum range) we fly airplane at speed corresponding to maximum L/D

– To stay aloft longest (maximum endurance) we fly the airplane at minimum PR or fly at a velocity where CL

3/2/CD is a maximum

• Power will also provide information on maximum rate of climb and altitude

27

POWER AVAILABLE AND MAXIMUM VELOCITY (6.6)

Propeller DriveEngine

1 hp = 550 ft lb/s = 746 W

PR

PA

28

POWER AVAILABLE AND MAXIMUM VELOCITY (6.6)

Jet Engine

P A = T A

V ∞

PR

29

ALTITUDE EFFECTS ON POWER REQUIRED AND AVAILABLE (6.7)

Recall PR = f(∞)Subscript ‘0’ denotes seal-level conditions

21

00,,

21

00

RALTR

ALT

PP

VV

30

ALTITUDE EFFECTS ON POWER REQUIRED AND AVAILABLE (6.7)Propeller-Driven Airplane

Vmax,ALT < Vmax,sea-level

31

RATE OF CLIMB (6.8)

• Boeing 777: Lift-Off Speed ~ 180 MPH

• How fast can it climb to a cruising altitude of 30,000 ft?

32

RATE OF CLIMB (6.8)

cos

sin

WL

WDT

Governing Equations:

33

RATE OF CLIMB (6.8)

sin/

sin

sin

sin

VCR

VW

DVTV

WVDVTV

WDT

Vertical velocity

Rate of Climb:

TV∞ is power availableDV∞ is level-flight power required (for small neglect W)TV∞- DV∞ is excess power

34

RATE OF CLIMB (6.8)

Jet EnginePropeller Drive Engine

Maximum R/C Occurs when Maximum Excess Power

35

EXAMPLE: F-15 K• Weapon launched from an F-15 fighter by a small two stage rocket, carries a heat-

seeking Miniature Homing Vehicle (MHV) which destroys target by direct impact at high speed (kinetic energy weapon)

• F-15 can bring ALMV under the ground track of its target, as opposed to a ground-based system, which must wait for a target satellite to overfly its launch site.

36

GLIDING FLIGHT (6.9)

DL

L

D

1tan

cos

sin

cos

sin

0

WL

WD

T

To maximize range, smallest occurs at (L/D)max

37

EXAMPLE: HIGH ASPECT RATIO GLIDER

To maximize range, smallest occurs at (L/D)max

A modern sailplane may have a glide ratio as high as 60:1So = tan-1(1/60) ~ 1°

RANGE AND ENDURANCE

How far can we fly?

How long can we stay aloft?

How do answers vary for propeller-driven vs. jet-engine?

39

RANGE AND ENDURANCE• Range: Total distance (measured with respect to the ground) traversed by airplane

on a single tank of fuel

• Endurance: Total time that airplane stays in air on a single tank of fuel

1. Parameters to maximize range are different from those that maximize endurance

2. Parameters are different for propeller-powered and jet-powered aircraft

• Fuel Consumption Definitions

– Propeller-Powered:

• Specific Fuel Consumption (SFC)

• Definition: Weight of fuel consumed per unit power per unit time

– Jet-Powered:

• Thrust Specific Fuel Consumption (TSFC)

• Definition: Weight of fuel consumed per unit thrust per unit time

40

PROPELLER-DRIVEN: RANGE AND ENDURANCE• SFC: Weight of fuel consumed per unit power per unit time

hourHP

fuel of lbSFC

• ENDURANCE: To stay in air for longest amount of time, use minimum number of pounds of fuel per hour

HPSFChour

fuel of lb

• Minimum lb of fuel per hour obtained with minimum HP

• Maximum endurance for a propeller-driven airplane occurs when airplane is flying at minimum power required

• Maximum endurance for a propeller-driven airplane occurs when airplane is flying at a velocity such that CL

3/2/CD is a maximized

41

PROPELLER-DRIVEN: RANGE AND ENDURANCE• SFC: Weight of fuel consumed per unit power per unit time

hourHP

fuel of lbSFC

• RANGE: To cover longest distance use minimum pounds of fuel per mile

V

HPSFC

mile

fuel of lb

• Minimum lb of fuel per hour obtained with minimum HP/V∞

• Maximum range for a propeller-driven airplane occurs when airplane is flying at a velocity such that CL/CD is a maximum

42

PROPELLER-DRIVEN: RANGE BREGUET FORMULA

• To maximize range:

– Largest propeller efficiency, – Lowest possible SFC

– Highest ratio of Winitial to Wfinal, which is obtained with the largest fuel weight

– Fly at maximum L/D

final

initial

D

L

W

W

C

C

SFCR ln

43

PROPELLER-DRIVEN: RANGE BREGUET FORMULA

final

initial

D

L

W

W

C

C

SFCR ln

PropulsionAerodynamics

Structures and Materials

44

PROPELLER-DRIVEN: ENDURACE BREGUET FORMULA

• To maximize endurance:

– Largest propeller efficiency, – Lowest possible SFC

– Largest fuel weight

– Fly at maximum CL3/2/CD

– Flight at sea level

2

12

12

123

2 initialfinalD

L WWSC

C

SFCE

45

JET-POWERED: RANGE AND ENDURANCE• TSFC: Weight of fuel consumed per thrust per unit time

hour thrustof lb

fuel of lbTSFC

• ENDURANCE: To stay in air for longest amount of time, use minimum number of pounds of fuel per hour

ThrustTSFChour

fuel of lb

• Minimum lb of fuel per hour obtained with minimum thrust

• Maximum endurance for a jet-powered airplane occurs when airplane is flying at minimum thrust required

• Maximum endurance for a jet-powered airplane occurs when airplane is flying at a velocity such that CL/CD is a maximum

46

JET-POWERED: RANGE AND ENDURANCE• TSFC: Weight of fuel consumed per unit power per unit time

hour thrustof lb

fuel of lbTSFC

• RANGE: To cover longest distance use minimum pounds of fuel per mile

V

ThrustSFC

mile

fuel of lb

• Minimum lb of fuel per hour obtained with minimum Thrust/V∞

• Maximum range for a jet-powered airplane occurs when airplane is flying at a velocity such that CL

1/2/CD is a maximum

D

L

DL

R

CC

CSC

WS

V

T

21

12

2

1

47

JET-POWERED: RANGE BREGUET FORMULA

• To maximize range:

– Minimum TSFC

– Maximum fuel weight

– Flight at maximum CL1/2/CD

– Fly at high altitudes

21

212

112

2 finalinitialD

L WWC

C

TSFCSR

48

JET-POWERED: ENDURACE BREGUET FORMULA

• To maximize endurance:

– Minimum TSFC

– Maximum fuel weight

– Flight at maximum L/D

final

initial

D

L

W

W

C

C

TSFCE ln

1

49

SUMMARY: ENDURANCE AND RANGE• Maximum Endurance

– Propeller-Driven• Maximum endurance for a propeller-driven airplane occurs when airplane is

flying at minimum power required• Maximum endurance for a propeller-driven airplane occurs when airplane is

flying at a velocity such that CL3/2/CD is a maximized

– Jet Engine-Driven• Maximum endurance for a jet-powered airplane occurs when airplane is flying

at minimum thrust required• Maximum endurance for a jet-powered airplane occurs when airplane is flying

at a velocity such that CL/CD is a maximum

• Maximum Range– Propeller-Driven

• Maximum range for a propeller-driven airplane occurs when airplane is flying at a velocity such that CL/CD is a maximum

– Jet Engine-Driven• Maximum range for a jet-powered airplane occurs when airplane is flying at a

velocity such that CL1/2/CD is a maximum

EXAMPLES OF DYNAMIC PERFORMANCE

Take-Off Distance

Turning Flight

51

TAKE-OFF AND LANDING ANALYSES (6.15)

F

mVs

Vdtds

tm

FV

dtm

FdV

dt

dVmmaF

2

2

dt

dVmLWDTRDTF r

Rolling resistancer = 0.02

s: lift-off distance

52

NUMERICAL SOLUTION FOR TAKE-OFF

53

USEFUL APPROXIMATION (T >> D, R)

• Lift-off distance very sensitive to weight, varies as W2

• Depends on ambient density

• Lift-off distance may be decreased:

– Increasing wing area, S

– Increasing CL,max

– Increasing thrust, T

TSCg

Ws

LOL

max,

2

..

44.1

sL.O.: lift-off distance

54

EXAMPLES OF GROUND EFFECT

55

TURNING FLIGHT

V

ng

R

V

dt

d

ng

VR

R

VmF

nWF

W

Ln

WLF

WL

r

r

r

1

1

1

cos

2

2

2

2

2

22

Load Factor

R: Turn Radius

: Turn Rate

56

EXAMPLE: PULL-UP MANEUVER

V

ng

ng

VR

R

VmF

nWWLF

r

r

1

1

1

2

2

R: Turn Radius

: Turn Rate

57

V-n DIAGRAMS

SWC

Vn

W

SCV

W

Ln

L

L

max,2max

2

2

1

21

58

STRUCTURAL LIMITS